In the mid '70s , I learned Ken Iverson'sAPL to
ease my learning the multidimensional geometry I needed to
grok papers which seemed to be saying something
non-trivial .
Since then , for me , to understand something is to be
able to compute it . So , I approach the understanding of
mean
planetary temperature from a very classical physical reductionist
perspective looking at how much of the unexplained delta between our
known physics and our observations is reduced by by each added
parameter .

Joe Bast invited me to start a page on their ClimateWiki taking
this essential
physics approach . So far , on that page , I work thru the
calculation of the temperature of an opaque uniform gray ball in our
orbit in
an evolute of Iverson's array programming language , Arthur Whitney's K
. Tables are set up for others to translate the quantitative
relationships into other programming notations . I use Arthur's K
here but the expressions are generally quite similar to their
traditional textbook equivalents . It's just that they operate on
entire lists . Martin Hertzberg proves more
simply than I, that the equilibrium temperature of a radiantly heated
colored ball is

where Tgray is the StefanBoltzmann calculated
temperature for a flat spectrum , ie , gray ,
body . The equilibrium temperature of a colored ball is the temperature
for a gray ball times the fourth root of the ratio
between its absorptivity and its emissivity
. For a flat-spectrum , that ratio is 1 ; so a
gray
ball
comes to the temperature calculated by simply adding up the energy
impinging on a point in our orbit as calculated on the essential
physics page regardless of how light or dark it is .

So how are absorptivity and emissivity for a uniformly colored , that
is non-flat-spectrum , ball computed ? At this point it becomes very
hard to find any explanation on the web .

It depends on the spectra of the radiant sources and sinks , and the
spectrum of the ball . Here is the computation simply using
Planck thermal radiation spectra :

First , we need to calculate the spectra of source , the sun , and that
of a neutral gray ball emitting the same energy in our orbit

The Planck
function , the crowning achievment of 19th century
physics , provides the
spectrum for a black ( totally absorptive-emissive ) body as a function
of temperature . Particularly for lower temperatures , it has very long
tails towards longer wavelengths . First
we create a list of wavelengths in the range of interest . ( The
definitions of the defined words are in a table below . )

We will use the commonly cited temperature of 5778 kelvin for the sun's
temperature , and the temperature of a gray ball in our orbit , 278.68
, calculated from that using the StefanBoltzmann law as we'll see in a
moment .

Spectra : pi * 1e-7 *
Planckl[ WL ]' 5778 278.68

This expression produces a 2 column table for power density for each of
the wavelengths in WL for each of the 2
temperatures . The 1e-7 scales
the power to the 0.1 micron increment . Note that Planckl
returns power per steradian while the total energy is emitted over a
hemisphere . Thus the factor of pi
.
We can test the accuracy of these curves and the extent to which our
range of wavelengths is sufficient to include all the energy by
comparing the sum across the Spectra to the total
energy calculated by the Stefan
Boltzmann law which pre-dates Planck's function .

( +/' Spectra ; T2Psb 5778 278.68 )

6.319533e+007

6.320098e+007

339.3557

342.0085

%/ r />/ 0.9999106
0.9922434
/ r contains the result of the last operation

So even going out to 90 microns , we're still off by about 0.8% on the
278k spectrum .
This may not seem like much , but the commonly agreed upon
total change in our temperature since before the industrial revolution
is from about 288 kelvin to 288.8 , only about 0.3% .
But all these numbers are not as precise as often portrayed . No Tricks Zone
has an interesting post showing
disagreement in our mean temperature of more than 2 degrees .
There are lots of ways to increase the precision of these
computations , but simple , first .

You can see above what an enormous difference in radiance that 4th
power StefanBoltzmann relationship creates . In fact , the sun radiates
more than 500 times as much as a gray ball in our orbit even at its
peak around 10 microns .

At any given wavelength , an opaque object may either
absorb
radiation of that wavelength converting it in to thermal motion or
reflect it .

( reflect + ae ) = 1
/ at each wavelength

Over 150 years ago , Kirchhoff
and others proved that since they were just two directions
thru the same process , absorptivity and emissivity
are always equal at each frequency . For a gray
flat-spectrum body , ae = constant , across all
wavelengths , clearly the constant
drops out which is why we can the temperature of the spectrum above is
that of a gray body , however light or dark , not just as frequently
asserted , a black body .

Note : I rather hate the term albedo
which is a synonym for reflectance . Its imprecise
usage causes much confusion . As we've asserted ,
reflectance or Albedo
, is a function of wavelength . It should either be expressed as albedo
with respect to some spectrum , or , unmodified , should refer to a
flat spectrum , in which case it drops out of the equation for
temperature .

As can be seen in the graph above , the solar spectrum
and the thermal spectrum for
a gray body in our orbit are almost disjoint . Therefore it 's
acceptable to split the spectrum into short-wave ( solar ) and
long-wave ( planetary ) portions . The crossover point can be
found by looking for the first place the earth spectrum is greater than
the solar :

Going
back to Hertzberg's expression , we can calculate the ratio between the
mean absorptivity over the sun's spectrum to the emissivity over the
planet's . Given the temperatures used above , 288 and 288.8 for the
observed , and 278.68 for Tgray ,
we get

( 288. 288.8 % 278.68 ) ^
4 />/ 1.140635 1.153362

The commonly used value for the lumped earth+atmosphere absorptivity with
respect to the sun's spectrum is
0.7 . So we can make a step function spectrum breaking at 4.4 microns ,
with an absorptivity of 0.7 over the short wavelengths , and

.7 % r />/ 0.6136932
0.6069215

We'll check just the first of these .

AEspectrum : ,/ ( 43 , 900
- 43 ) #' 0.7 , * r

The line below uses K's secant search to find the
temperature x whose spectrum weighted by AEspectrum
matches the area of the scaled sun's spectrum weighted by the same AEspectrum
.

The ubiquitous null
hypothesis
seemingly accepted by all sides is that the surface of the earth has
an ae
with respect to the sun of that 0.7 , but , an absolute black body ,
1.0 , in the longer wave lengths . That's the most extreme
assumption that can be made given the 0.7 short wave length ae
. Here's what we get :

Here are the Sun and planet spectra and the stepped ae
spectra which matches our observed temperature , the frozen
planet
assumption used to scare everbody , and a flat spectrum with
an ae of 0.7 . These step spectra have
been scaled up by a factor of 10 .

The
crudeness , extremity , and hand waving justification of the Alarmist
step function hypothetical surface spectrum strikes me as simply
retarded versus any other branch of applied physics . Surely a better
estimate of earth's mean surface spectrum is available ? Give
me
any spectrum , and I'll give you it's equilibrium temperature . The
spectrum of liquid water would be a good start since it dominates the
surface of the earth . The spectrum , and therefore equilibrium
temperature of snow would be another interesting one . Would "snowball
earth" be stable ? It's significant that the gray
temperature is about 4 degrees warmer than the one obvious tipping
point around , the freezing point of water .

With
these sorts of computations , the delta in temperature from a delta in
atmospheric CO2 should be straight forward to calculate .

But
the earth surface is far from a uniform color . Nor are
clouds
uniformly distributed . But it takes little more code to add a
Lambertian cosine function and individual spectra to any partition of
the sphere can be done just a succinctly as the computations above .

I
hope the computations here make it believable that a rather detailed
planetary model could be expressed in not more than a couple of pages
of succinct , and therefore transparent , definitions .

I
reserve the
right to post all communications I receive or generate to CoSy website
for further reflection . Contact
: Bob Armstrong
; About this page : Feedback
Woodland
Park , Colorado / 80863-9711
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